Zeitschrift for P h y s i k B
Z. PhysikB 25, 255-258 (1976)
© by Springer-Verlag 1976
Hall-Resistivity of Amorphous Ferromagnetic Ni--Au-Alioys G. Bergmann Institut ffir Festk~Srperforschung, Kernforschungsanlage Jfilich, Jtilich, W. Germany Received July 9, 1976 Amorphous films of NicAu~_ C with atomic concentrations 0.35 0.5. The anomalous Hall-resistivity is of the order of 10-Sf2m and negative, in contrast to the Hallresistivity of amorphous Fe- and Co-alloys. The high field Hall-effect and the high temperature Hall-effect are interpreted as being the normal Hall-effect. Within the framework of the free-electron model one obtains between 0.6 and 0.8 conduction electrons per atom for the different alloys.
§ 1. Introduction The Hall-effect in ferromagnetic metals shows rather interesting properties. Besides the normal Hall-effect which is caused by the Lorentz force, one also finds an anomalous or spontaneous Hall-effect, which is very large compared to the former one. It is due to the anisotropic scattering of the conduction electrons by the magnetic moments of the metal. Since the normal Hall-effect of nonferromagnetic metals is already rather complicated due to their electronic structure, i.e. the shape of their Fermi surfaces etc., this generally complicates the separation into the normal and the anomalous Hall-effect. We have therefore investigated ferromagnetic metals in which these complications are suppressed. In amorphous metals, the fine structure of the Fermi surface is smeared out by the small mean free path and generally the conduction electrons behave as free electrons. In such an amorphous system the normal Hall voltage is proportional to the applied magnetic field. This simplifies the separation of the anomalous Hall-effect from the normal one. Phenomenologically the Hall resistivity of a ferromagnetic metal can be written as a sum of the normal and anomalous Hall-effect, E~/
Prx- j -RoBz+RsJz, R 0 is the normal Hall constant, R s is the anomalous Hall constant, Bz is the applied magnetic field in z-
direction and Jz is the component of the magnetization in the z-direction.* This phenomenological description of the total Hall resistivity offers two possibilities to separate the normal and the anomalous Hall-effect. First it is possible to saturate the magnetization in the amorphous metal so that for sufficiently high fields the Hall resistivity is determined by the normal Halleffect. Secondly one can investigate the ferromagnetic metal above the Curie temperature, where J~=zB~ and the susceptibility obeys a Curie-Weiss law. By extrapolation to infinite temperature one obtains again the normal Hall constant. Such a measurement has been performed by the author [1] for the amorphous alloy Nio.65Auo.35. The two independently measured values for the normal Hall constant agreed quite well. In this paper we want to investigate the spontaneous Hall resistivity and the high field Hall constant (normal Hall constant) of NiAu-alloys as a function of concentration. We try to extract from the data the number of conduction electrons in these amorphous alloys. In particular we hope, that the data of the spontaneous Hail-effect contribute to a better understanding of the anomalous Hall-effect.
§ 2. Experiment and Results The amorphous ferromagnetic alloys are prepared as thin films by quenched condensation. The alloy * We use the mks-systemwithB=#oH+J
256
G. Bergmann: Hall-Resistivity of A m o r p h o u s Ferromagnetic N i - A u - A l l o y s Fig. 1. The Hall-resistivity of an a m o r p h o u s Nio.83Au0.17-alloy. The right scale is expanded
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is evaporated from aBe-oxidecrucible which is heated by means of a tungsten wire. The vapor is condensed onto a quartz plate held at the temperature of liquid helium. The vacuum during condensation is of the order of 10- 8 Torr. The thickness of the film varies between 1,000 and 1,500 A, and is measured by an interference method due to Tolansky and with a surface profile measuring system*. Since Ni and Au have different vapor pressures at the evaporation temperature, the concentration in the film differs from the original concentration of the alloy. We determine the concentration in the film by means of an X-ray fluorescence analysis to an accuracy of order 5 %; this is described in the Appendix. Generally the alloys show an irreversible decrease of the resistivity during annealing, which is typical for a transition from the amorphous to the crystalline state. For alloys in the concentration range 0 . 5 < c < 0 . 7 the crystallization temperature lies above room-temperature. Therefore we used an alloy of Ni0.65Au0.35 to investigate the structure by electron diffraction. We obtained the smeared pattern which is characteristic for a liquidlike amorphous structure. After condensation the evaporation cryostat is inserted into a superconducting magnet. Details of the apparatus and the procedure are described in Reference 2. A magnetic field up to 8 0 k G can be applied perpendicular to the film. In Figure 1 the Hall resistivity of an alloy Nio.83Auo.lv is plotted as a function of the applied field. Since the saturation magnetization of this alloy is rather small, a weak field is already sufficient to align the spins perpendicular to the film plane. A further increase of the applied magnetic field causes essentially a normal * F r o m Fa. Sloan
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-
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o/
- 0.5
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Fig. 2. The spontaneous Hall-resistivity as a function of the Niconcentration. Ps vanishes in the non-lerromagnetic alloys for c < 0.5
Hall voltage with negative slope. We denote the slope of the Hall resistivity in the high field range by Roo. The spontaneous Hall resistivity Ps is obtained by extrapolation to zero magnetic field. Its values are plotted in Figure 2 as a function of the Ni-concentration. Since the spontaneous Hall resistivity is essentially proportional to the magnetization it disappears when the amorphous alloy has lost its ferromagnetism. This takes place for Ni-concentration below 0.5 which are not ferromagnetic. In addition it allows the determination of the Curie temperature for higher concentrations [3]. The high field anomalous Hall constant is analysed in § 3 in terms of the number of conduction electrons per atom. In the Ni-concentration range between 0.50 and 0.65 we measured the high field Halleffect at He-temperature and the high-temperature Hall-coefficient extrapolated to T ~ or. Both values agree within 5 %-10 %.
G. Bergmann: Hall-Resistivity of Amorphous Ferromagnetic Ni-Au-Alloys
257
§ 3. Discussion
One obtains
a) The Anomalous Hail Resistivity
R ~ = Ro + ~
T h e anomalous Hall resistivity of NiAu alloys is negative. This is in clear contrast to amorphous Fe [-4, 51 and Co alloys [-6, 3] where the spontaneous Hall resistivity is postive. The dependence of the Hall resistivity on the Ni-concentration is rather weak at high Ni-concentrations. This means that the contribution per Ni atom increases in this range with decreasing Ni-concentration. In crystalline Ni the spontaneous Hall resistivity shows both signs depending on the defect structure of the material [-7]. There exist quite a large number of theories about the anomalous Hall-effect [-7]. We wish to refer to a paper by Fert and Jaoul [-8] who regarded the scattering of the conduction electrons by a magnetic transition atom in a phase shift treatment. They assumed resonance scattering at the d-states and a splitting of the resonance energies for different magnetic quantum numbers m. As a result of this calculation the Hall resistivity of spin up and down electrons is essentially proportional to s i n 2 6 f . The phase shift 6~- of the spin up electrons is in general close to ~. This applies for pure Ni, for Co and for (amorphous) Fe with more than 10 % Au. Therefore the spin up electrons hardly contribute to the anomalous Hall effect. The phase shift of the spin down electrons is given by 62 = z2 • z/5. Assuming one conduction electron per atom we find an increasing series for 6~ from Fe to Ni: 2 ~z/5, 3 n/5 and 4 re/5. Since the Hall effect is proportional to sin 2 62 it must change its sign in going from Fe to Ni. In the N i - Au system we expect with increasing Au concentration that 6 + and 6j approach each other and at 50 % Au both are equal and the resulting anomalous Hall effect vanishes. The advantage of the amorphous alloys is that the conduction electrons are considered as free. They are scattered by the total potential of the atoms [9] and not only by the small deviation from the periodic potential as in the crystalline case. Therefore the theoretical treatment of an amorphous alloy is in many respects similar to that of dilute or isolated atoms. One has only to include the structure factors of the alloys. The advantage is, however, that one can perform the measurements at very high concentrations.
Besides the normal Hall-constant we may have an additional term if the anomalous Hall-resistivity is field dependent. One possible reason may be a highfield susceptibility of the alloys. A comparison of the experimental results for Ni-alloys with those of Fe and Co-alloys does not support this possibility because all these alloys show a negative high-field Halleffect although their anomalous Hall-effect varies in sign and magnitude. The Hall constant for simple liquid and amorphous metals has not yet been successfully calculated. However, the experimental results yield surprisingly good agreement with a free electron model. Therefore we try to derive the number of conduction electrons z~ in amorphous NiAu alloys from the high field constant. In contrast to simple amorphous metals, the conduction electrons with spin up and spin down have different mean free paths and therefore contribute differently to the normal Hall constant. This is already due to the isotropic exchange scattering term which can be described by 2 - s S I w(R-Rz) [10]. In the ferromagnetic state the effective potential of spin up and down electrons is different, and given by
b) High-Field Hall Constant According to the above equation for the Hall resistivity in ferro-magnetic metals the high-field Hall coefficient R~ offers a possibility to obtain the normal Hall constant. However, the evaluation of R~ is not trivial.
(Rs "J=).
v(r - r~) +_ S w ( r - r~).
In the paramagnetic state, however, the mean free path of both conduction electrons are identical. Since we obtained for Ni0.65Au0.35 almost the same Hall constant in the ferromagnetic and the paramagnetic state, the influence of the spin-dependent scattering potential is rather small. However, for higher Ni concentrations this might be different. Generally, the effective number zeff of conduction electrons which one obtains from the normal Hall coefficient lies within Zs/2 < Zeff < Z s. The effective number of conduction electrons per atom is given by z ff=eleol ' where ~ is the average atomic volume in the alloy. We estimated ~=CNi~Ni-I-(1--CNi)~"~Au, taking the atomic volume from the crystalline phase, and assume a reduced density of about 10 % in the amorphous phase. This effective number of conduction electrons Zeff is drawn in Figure 3 as a function of the Ni concentration. The dashed line corresponds to the number which one expects, if each Au atom contributes one conduction electron per atom and each Ni atom 0.6 conduction electrons. A rigid-band model with 0.5 holes in the d-band of pure Ni is plotted in the dashed
258
G. Bergmann: Hall-Resistivity of Amorphous Ferromagnetic Ni-Au-Alloys
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Fig. 3. The number Zeff of effective conduction electrons per atom. Zeff is obtained from the high-field Hall-effect within the free electron model
Fig. 4. The Ni and Au fluorescence rate for different films. The numbers at the points describe the composition of the original alloys before the evaporation
dotted line. The experimental points show a strong scattering. They lie clearly beiow the dashed line.
The slope of the curve is given by /7/=---
Appendix. X-Ray Fluorescence Analysis For the determination of the composition of the evaporated films, we bring the films into an X-ray beam of given primary intensity. The intensity of the fluorescence spectrum is proportional to the number of atoms within this beam because, for film thicknesses of the order of 1,000 A, the absorption can be neglected. If Aau and ANi give the fluorescence rate for one Au or Ni atom respectively, then the fluorescence rate of the film is given by ni.A ~, where n~ is the number of atoms i in the effective cross section of the film F. This number is given by l~i=Ci
F.d
Since we used the same effective cross-section for all investigated films, we get a simple linear relation between the Au and the Ni rate divided by the thickness of the film.
ANi
~"~Au
AAu ~2Ni
With this slope we obtain the concentration in the films Cil
I'CNi
ZNi ~e~Ni tml ZAu
faAu 1
Experimentally it turns out that the Au-concentration in the films is increased relative to the concentration in the original alloy.
References 1. 2. 3. 4. 5. 6. 7.
Bergmann, G.: Solid State Commun. 18, 897 (1976) Bergmann, G.: Phys. Rev. B7, 11 (1973) Bergmann, G.: to be published Lin, S.C.H.: J. Appl. Phys. 40, 2175 (1969) Marquardt, P., Bergmann, G.: to be published Whyman, P., Aldrige, R.V.: J. Phys. F: Metal Phys. 4, L6 (1974) Proceedings of the Meeting: L'effect Hall extraordinaire. Sait-Cergue 1972, ed. G.Cohen, B.Giovannini and O.Sorg. 8. Fert, A., Jaoul, O.: Phys. Rev. Lett. 28, 303 (1972) 9. Ziman, J.M.: Phil. Mag. 6, 1013 (1961) 10. Kasuya, T.: Progr. Theor. Phys. 16, 58 (1956)
ZNi ~'~Ni ZAu £r2Au d ANi
d
AAu
In Figure 4 this linear relation is confirmed by the experimental points. The numbers at the points give the composition of the original alloys. The advantage of this method with thin films is that it calibrates itself.
Bergmann Institut ftir FestkSrperforschung Kernforsc'gungsanlage Jiilich GmbH Postfach 1913 D-5170 Jiilich Federal Republic of Germany
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